I own several motorcycles, and a lot of people I know ride motorcycles. Most of them are safe and responsible. If the motorcycle driver is responsible, then the level of threat isn’t as high as the neighbor in the OP thinks.
That said, if someone didn’t want to ride on the back of my bike, I wouldn’t be offended. Mrs. Geek has no desire to get on the back of a motorcycle. No biggie.
Some people think that motorcycles are death machines and that no sane person should ever ride one. They aren’t anywhere near that dangerous. Is the OP’s neighbor overreacting? A bit, but like I said, no biggie. You probably aren’t going to change their opinion no matter what you say.
This is the big one. Car drivers are usually looking for cars, and a motorcycle isn’t a car. I’ve had people look right at me at an intersection, and then proceed to completely ignore me and cut me off. I have also had people try to merge into me, not realizing I was there. As a motorcyclist, you almost have to be completely paranoid and assume that everyone on the road is out to get you. Paranoid or not, that kind of attitude can save your life.
Um… you’re supposed to lean into curves. How much were you leaning?
If you are riding on the back of the motorcycle, you should do these:
Lean into curves. Lean roughly as much as the motorcycle leans, which will be roughly as much as the driver leans. If the driver is taking turns so fast and leaning so much that their knee nearly hits the ground, you probably don’t want to be riding on the back of their bike. Carrying a passenger and racing around a track are two completely different things and should never be combined.
Keep your feet on the pegs, even when the motorcycle stops. The driver should tell you where to put your feet and what to avoid. The exhaust pipes get very, very hot. Also, make sure you don’t have long shoe laces that are dangling and might get caught in the chain/belt/shaft (whatever drives the rear wheel). A typical bike will have the chain/belt/whatever on the left and the exhaust pipes on the right, but some bikes have exhaust pipes on both sides.
The motorcycle might have a strap across the seat for the passenger to hang onto. Don’t use it. Hang on to the driver, not the strap. If you don’t want to put your arms around the driver, tough noogies. That’s what you need to do, whether you want to or not.
Don’t hang on to the driver’s arms. He/she needs those to be free to control the bike.
Wear a helmet. If they give you a full-face helmet to wear, that’s a good thing. Hitting a large bug with your face at 65 mph hurts. If they give you an open face helmet, make sure they also give you eye protection. Ideally you should also have a nice thick leather jacket and gloves. The chance of crashing is very small (assuming a responsible driver) but in the unlikely event that you do crash, the more leather you have the less skin you lose.
If the driver just tosses you a helmet and doesn’t tell you where to put your feet, how to lean, etc., that’s a bad sign.